I present a peer effects model for count data using a static game of incomplete information. I provide sufficient conditions under which the game equilibrium is unique. I estimate the model's parameters using the Nested Partial Likelihood approach and establish asymptotic properties of the estimator. I show that using the standard linear-in-means spatial autoregressive (SAR) model or the SAR Tobit model to estimate peer effects on counting variables generated from the game asymptotically underestimates the peer effects. I use the model to study peer effects on students participation in extracurricular activities, controlling for network endogeneity.